Let's break down each part of your question regarding the quadratic function \( h(t) = 6t^2 - 24t + 30 \).
Part A: Graph this function
You can use the Desmos calculator to input the function. You will need to go to Desmos Graphing Calculator and enter the function \( h(t) = 6t^2 - 24t + 30 \). This will give you the graph of the hawk's height over time.
Part B: What is the initial height of the hawk?
The initial height of the hawk occurs when \( t = 0 \). To find this, substitute \( t = 0 \) into the function:
\[ h(0) = 6(0)^2 - 24(0) + 30 = 30 \]
So the initial height is \( 30 \) feet. As an ordered pair, this is: \[ (0, 30) \]
Part C: At which point is the hawk at its lowest height?
The lowest point of a quadratic function (a parabola that opens upwards) can be found using the vertex formula. The time at which the vertex occurs is given by:
\[ t = -\frac{b}{2a} \]
where \( a = 6 \) and \( b = -24 \).
Calculating: \[ t = -\frac{-24}{2 \cdot 6} = \frac{24}{12} = 2 \]
Now, plug \( t = 2 \) back into the height function to find the height:
\[ h(2) = 6(2)^2 - 24(2) + 30 = 6 \cdot 4 - 48 + 30 = 24 - 48 + 30 = 6 \]
Thus, the lowest height occurs at: \[ (2, 6) \]
Part D: The hawk will be at its lowest point at time ___ seconds at a height of ___ feet.
From our calculations, the hawk will be at its lowest point at:
- Time: \( 2 \) seconds
- Height: \( 6 \) feet
So, you can summarize your answer as: The hawk will be at its lowest point at time 2 seconds at a height of 6 feet.
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